^{1}

^{1}

^{2}

^{1}

^{*}

A simple and elegant method to simulate single order reflection profile based on 1-D paracrystalline model has been proposed here. For variety of polymer films this approach has been applied to compute microcrystalline parameters like crystallite size and lattice strain. Other metallic oxide compounds are also analysed using this approach. Employing this model, X-ray diffraction patterns from various polymer samples have been analysed and corresponding microstructure parameters have been reported in this article.

Broadening of Bragg reflections in polymer/metal oxide samples has drawn attention of many crystallographers for developing a technique to quantify the patterns [

Let us consider, at random, a set of rods of length “d” such that this length is distributed around a mean value “x”. Here let us assume that the distance probability is a Gaussianin shape and it is identified by H_{1}(x). The average length is given by

Using H_{1}(x) we can calculate the probability distribution of length vectors by the propagation of errors. The position of second Gaussian is

where “*” represents convolution process. The average length of second vector is given the Equation (1) with H_{2}(x). The probability distribution thus produced forms a one dimensional lattice with an atom/molecule at lattice edge. The Gaussian function is normalised so that all the peaks have an area of 1. We can continue this to find nth lattice point. This is done as follows:

where_{1} is given by

With a substitution

Extending this to the nth Gaussian we get

Taking the Fourier cosine transformation, we have

the intensity profile due to convolution of “n” Gaussians.

Convolution of Gaussians and their Fourier transform: N (number of convoluted Gaussians) and g is percentage of lattice distortion. First Gaussian Peak is located at 0.4 nm and corresponding peak position in Fourier transformed is located at 15.70

Flowchart for the computation of microstructural parameters and simulation of the experiment pattern

Solution to the Equation (7) is

where

with

The crystallite size distribution at t shifted by a distance equal to dth position (last Gaussian), we have

where

Here (t − x) is the shape function. Thus the resulting shape function will be

We get

The Fourier transform of this equation is given by

Substituting for Q(x) we get

Consider the first term. Since p(t) is +ve, we take x → |x| and then carry out the integration by breaking it into two terms.

The first and second integrations give

With this incorporation of correction we have an expression for the intensity as

For computation of crystallite size and strain from X-ray diffraction pattern, it is important to have good experimental data. We have used eight polymer samples wherein reasonably good X-ray data is available. The samples are two cotton fibers (SAHANA, JAYADHAR), two silk fibers (TASSAR, MUGA) and two polymers (HPMC, PVA). The flowchart for the computation of crystallite size and strain is given in

. Microstructural parameters for natural and man-made polymers using the present and earlier methods

Samples | Peaks | d in nm | D in nm | Strain (%) | D in nm reported | Strain (%) values |
---|---|---|---|---|---|---|

PVA | 1 | 0.4544 | 1.36 ± 0.11 | 10.0 ± 0.8 | 2.92^{(a)} | 1.0 |

2 | 0.2267 | 0.23 ± 0.02 | 10.0 ± 0.8 | |||

HPMC | 1 | 0.9528 | 3.81 ± 0.30 | 10.0 ± 0.8 | 7.32^{(b)} | 1.0 |

2 | 0.4492 | 1.35 ± 0.11 | 22.2 ± 1.6 | 2.45 | 0.5 | |

TASSAR | 1 | 0.5323 | 1.60 ± 0.13 | 11.1 ± 0.9 | 3.0^{(c)} | 14.0 |

2 | 0.4389 | 1.76 ± 0.14 | 11.8 ± 0.8 | 2.70 | 9.0 | |

3 | 0.2678 | 0.54 ± 0.04 | 7.7 ± 0.6 | 1.60 | 4.0 | |

MUGA | 1 | 0.5274 | 2.11 ± 0.17 | 11.1 ± 0.9 | 3.10^{(d)} | 30.0 |

2 | 0.4389 | 1.32 ± 0.10 | 9.1 ± 0.7 | 2.20 | 30.0 | |

3 | 0.2654 | 0.53 ± 0.04 | 6.3 ± 0.5 | 1.10 | 5.0 | |

SAHANA | 1 | 0.5822 | 1.75 ± 0.14 | 10.0 ± 0.8 | 2.34^{(e)} | 0.1 |

2 | 0.5352 | 1.61 ± 0.13 | 8.3 ± 0.6 | |||

3 | 0.3857 | 1.54 ± 0.12 | 7.7 ± 0.6 | 3.48 | 0.1 | |

4 | 0.2584 | 0.52 ± 0.07 | 7.1 ± 0.5 | 3.16 | 0.2 | |

JAYADHAR | 1 | 0.5704 | 2.28 ± 0.17 | 10.0 ± 0.8 | 2.46^{(f)} | 0.1 |

2 | 0.5377 | 2.15 ± 0.17 | 11.1 ± 0.8 | |||

3 | 0.3870 | 1.16 ± 0.09 | 6.7 ± 0.6 | 3.37 | 0.1 | |

4 | 0.2562 | 0.51 ± 0.04 | 8.3 ± 0.7 | 2.16 | 2.0 |

^{(a)}[^{(b)}[^{(c)}[^{(d)}[^{(e)}[^{(f)}[

Comparison of experimental and simulated whole powder pattern X-ray profiles. Right side gives column length exponential distribution

Comparison of experimental and simulated whole powder pattern X-ray profiles. Right side gives column length exponential distribution

Samples chosen here represents reasonably amorphous in nature, in a sense that a well defined delta-type Bragg reflections are rarely observed. Scherrer and Williamson-Hall methods relay on 1) instrumental broadening correction, 2) background correction and 3) accuracy of measuring FWHM using peak fit and peak separation procedures. Multiple and single order Fourier methods by Warren and others do relay on initial Fourier coefficients. In all these methods, our experience suggest that while refining the parameters against the profile and hence the whole powder pattern, the program selectively choose to give more weightage to the crystalline parameter than lattice strain parameter leading to either over or under estimation of crystallite size. Normally in polymers we have a situation where in a few unit cells contributes for a Bragg reflection. Under these circumstances we look for a method which can compute microstructural parameters with reasonable accuracy in a straight forward manner. In this direction, present method is quite useful. As mentioned in the flow chart, an approximate crystallite size is estimated using Nandi et al.’s initial slope method for a selected profile after all the corrections. This is done using Graphic User Interface technique wherein the beginning, peak and end of the profile is selected graphically in the program. Then the parameters like background, size and strain are varied continuously such that the final parameters are used to simulate the profile. This process is continued for all the available profiles. At the end of the day, all the profile parameters are used to compute the whole powder pattern to an accuracy which is always less than 5 percent between the simulated and experimental profiles, which is also verified visually using the graphics user interface in the program. The results are in broad agreement with single order Fourier method based on Warrens approach used earlier.

Essentially, crystallite size is the region between defects. Associated with this, there are non-uniform strains leading to shifts of atoms/molecules from their ideal positions. These two together along with other extended defects lead to peak broadening. Absolute accuracy of these results is always a problem. The restrictions are less stringent if one seeks results in a semi-quantitative manner. Here in a straight forward method, using exponential column length distribution, crystallite size is computed by a visual inspection of the profile and also the whole pattern. It is observed that it is difficult to extract accurate parameters from Rietveld analysis when the peak shape is mixed Gaussian-Lorentzian. Here the accuracy can be improved by the goodness off it in terms of simplified peak shape.

Authors acknowledge UPE and CPEPA grants from UGC, NewDelhi, India.

Program is freely available at the site faculty.physics.uni-mysore.ac.in/rs.